Ergodic Theory and Statistical Mechanics
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1115 Jean Moulin Ollagnier
Erqodic Theory and Statistical Mechanics
Spri nqer-Verlaq Berlin Heidelberg New York Tokyo
Author
Jean Moulin Ollagnier Departernent de Mathematiques, Universite Paris Nord Avenue J. B. Clement, 93430 Villetaneuse, France
AMS Subject Classification (1980): 20F, 28D, 54H20, 82A05, 82A25 ISBN 3-540-15192-3 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15192-3 Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Oftsetdruck, Hemsbach/Bergstr. 2146/3140-543210
CONTENTS
v
INTRODUCTION 1. PRELIMINARY ANALYSIS 1.1. Sublinear functions and the Hahn-Banach theorem 1.2. Compact convex sets 1.3. Radon measures 1.4. Extremal points in compact convex sets 1.5. References
. . . . . .
2. DYNAMICAL SYSTEMS AND AMENABLE GROUPS 2.1. Dynamical sys tems 2.2. The fixed point property and the ameaning filter 2.3. Amenability and algebraic constructions '" 2.4. References
. .
1 4
6
10 14 15 15
19
. .
3. ERGODIC THEOREMS . 3.1. Invariant linear functionals . 3.2. Invariant vectors and mean ergodic theorems . 3.3. Individual ergodic theorems . 3.4. The saddle ergodic theorem . 3.5. References ........................................................................................ 4. ENTROPY OF ABSTRACT DYNAMICAL SYSTEMS 4.1. Equivalence of abstract dynamical systems .. , 4.2. Entropy of partitions 4.3. Entropy of dynamical systems 4.4. The almost subadditive ergodic theorem and the Shannon-McMillan theorem 4.5. References
1
30 33
35 35 41
47 50
52
. . . .
53
. .
64
53 54 59
70
IV
5. ENTROPY AS A FUNCTION AND THE VARIATIONAL PRINCIPLE 5.1. Topological entropy
71 71
5.2. Pressure of a continuous function and the variational principle 72 5.3. Entropy as a function of the measure
..
82
5.4. References
85
6. STATISTICAL MECHANICS ON A LATTICE ..
86
6.1. Local specifications and Gibbs measures
86
6.2. Cocycles and quasi-invariant measures ...............•
91
6.3. Phase transitions....................................
96
6.4. Supermodular interactions
98
6.5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . 7. DYNAMICAL SYSTEMS IN STATISTICAL MECHANICS
104 105
7.1. Invariant local specifications
105
7.2. Invariant Gibbs measures and equilibrium measures
106
7.3. Mixing properties
108
7.4. Example: a theorem of Ruelle's
112
7.5. References
115
8. EQUIVALENCE OF COUNTABLE AMENABLE GROUPS ..
117
8.1. Tiling amenable groups
117
8.2. Equivalence of countable groups.
124
8.3. Rokhl in' s
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